Anomalous dimensions at finite conformal spin from OPE inversion

TL;DR

This paper calculates anomalous dimensions of higher spin operators in Conformal Field Theory across arbitrary dimensions using the OPE inversion formula, employing Mellin space techniques for efficiency and generality.

Contribution

It introduces a Mellin space approach to compute anomalous dimensions at finite conformal spin, providing exact results for scalar and spin exchanges in any dimension.

Findings

Exact expressions for anomalous dimensions using hypergeometric functions.

Reproduction of known large spin results as special cases.

Demonstration of Mellin space advantages over position space methods.

Abstract

We compute anomalous dimensions of higher spin operators in Conformal Field Theory at arbitrary space-time dimension by using the OPE inversion formula of \cite{Caron-Huot:2017vep}, both from the position space representation as well as from the integral (viz. Mellin) representation of the conformal blocks. The Mellin space is advantageous over the position space not only in allowing to write expressions agnostic to the space-time dimension, but also in that it replaces tedious recursion relations in terms of simple sums which are easy to perform. We evaluate the contributions of scalar and spin exchanges in the $t-$channel exactly, in terms of higher order Hypergeometric functions. These relate to a particular exchange of conformal spin $\beta=\Delta+J$ in the $s-$channel through the inversion formula. Our exact results reproduce the special cases for large spin anomalous dimension and OPE coefficients obtained previously in the literature.